First Fix the following notation:

$\forall \kappa\in Card~~~Tp(\kappa):="\kappa~has~tree~property"$

The large cardinals as "monsters of heaven" live everywhere in the land of mathematics. Most of them appear in topology, measure theory, infinitary logic, category, model theory, etc. But some of them live in a strange misty place, "the forest of tall trees"! One of the most important problems of set theory is to discover that which one of these monsters have a forestial nature? The forestial nature of a monster gives us very important information about his possible behavior at many combinatorial situations in set theory and model theory. Now define the following concepts:

**Definition (1):** Let $A$, $B$ be large cardinal axioms, $I$, $J$ be sub classes of $Ord$ , $\lbrace \alpha_{i}\rbrace_{i\in I}, \lbrace \beta_{j}\rbrace_{j\in J}\subseteq Ord$ , $\lbrace \alpha_{i}\rbrace_{i\in I}\cap \lbrace \beta_{j}\rbrace_{j\in J}=\emptyset$

then respectively every statement in the form $(\star)$ and $(\star\star)$ called a "weak" and "strong" tree property "equation". $(\star)~~Con(ZFC+A+\bigwedge_{i\in I}Tp(\aleph_{\alpha_{i}}))\Longleftrightarrow Con(ZFC+B+\bigwedge_{j\in J}Tp(\aleph_{\beta_{j}}))$

$(\star\star)~~(A+\bigwedge_{i\in I}Tp(\aleph_{\alpha_{i}}))\Longleftrightarrow (B+\bigwedge_{j\in J}Tp(\aleph_{\beta_{j}}))$

**Example (1):** Note to the following well known results:

A weak tree property equation: $Con(ZFC+Tp(\aleph_{2}))\Longleftrightarrow Con(ZFC+\exists~a~weakly~compact~cardinal)$

A strong tree property equation:

$(\kappa~is~strongly~inaccessible~+~Tp(\kappa))\Longleftrightarrow (\kappa~is~weakly~compact)$

**Remark (1):** It seems there are many weak tree property "unequalities" like the following results, but strangely the weak and strong "equalities" are too rare. These equalities uncover some fundamental relations between large cardinal axioms and combinatorial properties of other cardinals and so could be very useful and valuable.

A weak tree property unequality discoverd by Menachem Magidor: $Con(ZFC+ \exists~\mathtt{0}^{\sharp})\Longleftarrow Con(ZFC+Tp(\aleph_{2})+Tp(\aleph_{3}))$

Another weak tree property unequality discoverd by Uri Abraham: $Con(ZFC+\exists~a~supercompact~cardinal~with~a~weakly~compact~cardinal~above~it) \Longrightarrow Con(ZFC+Tp(\aleph_{2})+Tp(\aleph_{3}))$

**Question (1):** Is there any known large cardinal axiom stronger than "$\mathtt{0}^{\sharp}$ exists" and weaker than "there is a supercompact cardinal with a weakly compact cardinal above it " like $A$ in which we have an "weak tree property equality" in Magidor and Abraham's results? What about "strong tree property equality"? In the other words: is there any large cardinal axioms $A$, $B$, $C$ which the following statements be true?

$(1)~Con(ZFC+A)\Longleftrightarrow Con(ZFC+Tp(\aleph_{2})+Tp(\aleph_{3}))$ $(2)~B \Longleftrightarrow (C+Tp(\aleph_{2})+Tp(\aleph_{3}))$

**Question (2):** Are there any other known weak or strong tree property equality different from statements in example $(1)$?